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The Hoàng–Reed Conjecture states that a digraph with minimum out-degree d contains d dicycles C1,C2,…,Cd such that Ck intersects ⋃i<kCi on at most one vertex, for each k. It was made as a more structural approach to the Caccetta–Häggkvist Conjecture. We introduce the concept of a nearest separator and use it to prove a stronger version of the Hoàng–Reed Conjecture for the δ+=2 case. With these...
Let 𝒯 be the class of unlabeled trees. An unlabeled vertex‐deleted subgraph of a tree T is called a card. A collection of cards is called a deck. We say that the tree T has a deck D if each card in D can be obtained by deleting distinct vertices of T. If T is the only unlabeled tree that has the deck D, we say that T is 𝒯‐reconstructible from D. We want to know how large of a deck D is necessary...
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